Abstract
We study the power of constant-depth circuits containing negation gates, unbounded fan-in AND and OR gates, and a small number of MAJORITY gates. It is easy to show that a depth 2 circuit of sizeO(n) (wheren is the number of inputs) containingO(n) MAJORITY gates can determine whether the sum of the input bits is divisible byk, for any fixedk>1, whereas it is known that this requires exponentialsize circuits if we have no MAJORITY gates. Our main result is that a constant-depth circuit of size\(2^{n^{o(1)} } \) containingn o(1) MAJORITY gates cannot determine if the sum of the input bits is divisible byk; moreover, such a circuit must give the wrong answer on a constant fraction of the inputs. This result was previously known only fork=2. We prove this by obtaining an approximate representation of the behavior of constant-depth circuits by multivariate complex polynomials.
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References
J. Aspnes, R. Beigel, M. Furst, and S. Rudich, The expressive power of voting polynomials. InProc. Twenty-third Ann. ACM Symp. Theor. Comput., 1991, 402–409.
D. A. M. Barrington and H. Straubing, Complex polynomials and circuit lower bounds for modular counting. InProceedings of LATIN '92 (1st Latin American Symposium on Theoretical Informatics), 1992, 24–31.
D. A. M. Barrington, H. Straubing, andD. Thérien, Nonuniform automata over groups.Inform. and Comput. 89 (1990), 109–132.
R. Beigel, When do extra majority gates help? Polylog(n) majority gates are equivalent to one. InProc. Twenty-fourth Ann. ACM Symp. Theor. Comput., 1992, 450–454.
R. Beigel, N. Reingold, and D. Spielman, The perceptron strikes back. InStructure in Complexity Theory: Sixth Annual Conference, 1991a, 286–293.
R. Beigel, N. Reingold, and D. Spielman, PP is closed under intersection. InProc. Twenty-third Ann. ACM Symp. Theor. of Comput., 1991b, 1–9.
F. Green, An oracle separating ⊺P fromPP PH. InStructure in Complexity Theory: Fifth Annual Conference, 1990, 295–298.
N. Linial, Y. Mansour and N. Nisan, Constant-depth circuits, Fourier transforms and learnability. InProc. 30th Ann. IEEE Symp. Found. Comput. Sci., 1989, 574–579.
M. L. Minsky andS. A. Papert,Perceptrons. MIT Press, Cambridge, MA, 1988. (Expanded edition, original edition was in 1968.)
R. Smolensky, Algebraic methods in the theory of lower bounds for Boolean circuit complexity. InProc. Nineteenth Ann. ACM Symp. Theor. of Comput., 1987, 77–82.
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Barrington, D.A.M., Straubing, H. Complex polynomials and circuit lower bounds for modular counting. Comput Complexity 4, 325–338 (1994). https://doi.org/10.1007/BF01263421
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DOI: https://doi.org/10.1007/BF01263421